Environmental Engineering Reference
In-Depth Information
We now move to consider the case of a general
τ
(
P
)
. Consider a point
P
0
on
S
with
τ
0
=
τ
(
P
0
)
and
cos
(
PM
,
n
)
u
(
M
)
−
u
1
(
M
)=
[
τ
(
P
)
−
τ
(
P
0
)]
d
S
.
r
PM
S
=
The integral is uniformly convergent at
M
P
0
. For a surface
S
δ
containing
P
0
on
Ляпунов
S
, we have, using condition (4) for a
surface,
cos
(
PM
,
n
)
d
S
≤
k
(bounded)
.
r
PM
S
δ
Thus
(
,
)
cos
PM
n
[
τ
(
P
)
−
τ
(
P
0
)]
d
S
r
PM
S
δ
cos
(
PM
,
n
)
≤
max
P
δ
|
τ
(
P
)
−
τ
(
P
0
)
|
d
S
r
PM
∈
S
S
δ
≤
k
max
P
δ
|
τ
(
P
)
−
τ
(
P
0
)
| .
∈
S
By the continuity of
τ
(
P
)
, we thus have for any
ε
>
0andall
M
in a neighboring
region
V
of
P
0
,
cos
(
PM
,
n
)
[
τ
(
P
)
−
τ
(
P
0
)]
d
S
<
ε
r
PM
S
δ
provided that
S
δ
is sufficiently small such that all points on
S
δ
are sufficiently close
to
P
0
. Therefore,
u
(
M
)
−
u
1
(
M
)
converges at
P
0
uniformly. By Theorem 1,
u
(
M
)
−
u
1
(
M
)
must be continuous at
P
0
. By the definition of continuity, we have
u
(
P
0
)
−
u
1
(
P
0
)=
u
(
P
0
)
−
u
1
(
P
0
)=
u
(
P
0
)
−
u
1
(
P
0
)
.
Therefore, by Eq. (7.152),
u
(
P
0
)=
u
(
P
0
)+
u
1
(
P
0
)
−
u
1
(
P
0
)=
u
(
P
0
)
−
2
πτ
(
P
0
)
,
u
(
P
0
)=
u
(
P
0
)+
u
1
(
P
0
)
−
u
1
(
P
0
)=
u
(
P
0
)+
2
πτ
(
P
0
)
.
Since
P
0
is an arbitrary point on
S
, we arrive at Eq. (7.151).
This theorem is also valid for two-dimensional cases. However 2
π
in Eq. (7.151)
should be replaced by
π
.
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